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 A320067 Expansion of Product_{k>0} theta_3(q^k), where theta_3() is the Jacobi theta function. 21
 1, 2, 2, 6, 8, 10, 22, 26, 36, 60, 78, 106, 152, 202, 258, 370, 478, 602, 828, 1042, 1332, 1758, 2198, 2758, 3572, 4448, 5518, 7012, 8636, 10654, 13350, 16362, 19946, 24722, 30070, 36478, 44776, 54010, 65202, 79234, 95196, 114166, 137686, 164530, 196252, 235308, 279718, 332002 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also the number of integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama) Eric Weisstein's World of Mathematics, Jacobi Theta Functions FORMULA Expansion of Product_{k>0} eta(q^(2*k))^5 / (eta(q^k)*eta(q^(4*k)))^2. a(n) ~ log(2)^(3/8) * exp(Pi*sqrt(n*log(2))) / (4 * Pi^(1/4) * n^(7/8)). - Vaclav Kotesovec, Oct 05 2018 Expansion of Product_{k>0} theta_4(q^(2*k))/theta_4(q^(2*k-1)), where theta_4() is the Jacobi theta function. - Seiichi Manyama, Oct 26 2018 MATHEMATICA nmax = 50; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *) nmax = 50; CoefficientList[Series[Product[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *) PROG (PARI) m=50; x='x+O('x^m); Vec(1/(prod(k=1, 2*m, prod(j=1, floor(2*m/k), (1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2 )))) \\ G. C. Greubel, Oct 29 2018 (MAGMA) m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2: j in [1..Floor(2*m/k)]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018 CROSSREFS Cf. A000122, A029594, A033715, A320068, A320078, A320139, A320968, A320992. Sequence in context: A320246 A320247 A320248 * A248823 A284616 A136513 Adjacent sequences:  A320064 A320065 A320066 * A320068 A320069 A320070 KEYWORD nonn,nice AUTHOR Seiichi Manyama, Oct 05 2018 STATUS approved

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Last modified February 22 04:58 EST 2020. Contains 332115 sequences. (Running on oeis4.)